Dynamical vertex correction to the generalized Kadanoff-Baym Ansatz

The Non-Equilibrium Green's Functions (NEGF) are often used to formulate the fundamental equations of motion for highly perturbed many-body systems [1]. Less common is their direct numerical treatment since it represents a formidable task [2–11]. In order to simplify the NEGF equations of motion, an effective way is to approximate the double-time Green's function $G^<(t,t')$ by a functional of its time-diagonal element, the density matrix $\rho(t)=-{\rm i}\hbar G^<(t,t)$. One of such approximations is the Generalized Kadanoff-Baym Ansatz (GKBA) [12], $G^<(t,t^\prime) \approx G_{GKBA}^<(t,t^\prime)=\rho(t)G^A(t,t^\prime)-G^R(t,t^\prime)\rho(t^\prime)$, by which the NEGF equation of motion simplifies to the Generalized Master Equation (GME) for ρ. Recently, an effort is increasing to develop simple approximations beyond the GKBA [13–19].

The GKBA is the zeroth order of an expansion, $G^<=G_{ GKBA}^<+{\cal F}^<$, which includes vertex corrections ¹  ${\cal F}^<$, which are as complicated as the NEGF equation of motion. For their prohibitive complexity, there have been no attempts to include these vertex corrections into the GME, in spite of many implementations of the GKBA. Here we show that already a very simple approximation of vertex corrections can appreciably improve properties of the GME with only a minor increase in the numerical cost. We test this approximation on a molecular bridge between ferromagnetic leads. This system is sufficiently simple to be solved directly within the NEGF, at the same time, its dynamics is so complex that the GKBA fails.

In a previous paper [13] we have discussed the need for vertex corrections and tested a possibility to use the difference between the steady-state solution and the GKBA, ${\cal F}_\infty^<(t-t')=G_\infty^<(t-t')-\rho_\infty G^A(t-t^\prime)+G^R(t-t^\prime)\rho_\infty$, to this end. The corrected GME resulting from the corrected GKBA, $G^<(t,t^\prime) \approx G_{ GKBA}^<(t,t^\prime)+{\cal F}_\infty^<(t-t')$, converges to the known exact steady-state value, $\rho(t)\to\rho_\infty$, while the GME approaches a limit which deviates from $\rho_\infty$ by tens of %, see figs. 1 and 2.

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Although the correction ${\cal F}_\infty^<$ is vital, it has two drawbacks. First, its application is restricted to problems of relaxation towards a known steady state. Second, the vertices evolve in time while the function ${\cal F}_\infty^<(t,t')$ is independent of the center-of-mass time $(t+t')/2$. Here, we use the NEGF to derive an approximation of ${\cal F}^<$ which covers full “long-time” dynamics, while the “short-time” vertex is in the steady-state approximation. In comparison with the exact solution, it will be shown that such dynamically corrected GME predicts the time evolution of $\rho(t)$ with better accuracy than the corrected GME. The additional numerical cost is minor (less then 10% in CPU for our study).

Let us remind the basic features of the model of a molecular bridge and the description of its time evolution with the NEGF. The bridge has only a single quantum state connected to two leads. Non-trivial dynamics of this simple model follows from the complex spin-dependent energy spectrum of leads which corresponds to the surface density of states of nickel, see the appendix for details. Tunneling through the bridge with the energy level between Fermi levels of leads combines a fast dynamics of spin ↓ electrons and a slow dynamics of spin ↑ electrons. Simple approximations based exclusively either on the short-time or long-time evolution thus fail to describe the coupled motion of both components.

The $G^<(t,t^\prime)$ is a projection on the bridge site. We do not assume spin flips, therefore the GF has only spin-diagonal elements $G_\uparrow$ and $G_\downarrow$. The NEGF equation of motion,

$$\begin{align} &-{\rm i}\hbar\partial_{t'}G_\sigma^<(t,t')=G_\sigma^<(t,t') \varepsilon_\sigma(t')\nonumber\\ &+\int_{t_0}^{t'}{{\rm d} \bar{t} }\,G_\sigma^<(t,\bar{t})\Sigma_\sigma^A (\bar{t},t')+\int_{t_0}^{t}{{\rm d} \bar{t} }\,G_\sigma^R(t,\bar{t})\Sigma_\sigma^<(\bar{t},t'), \end{align} \tag{ 1 }$$

is complemented by the Dyson equation

$$\begin{align} {\rm i}\hbar\partial_tG_\sigma^R(t,t') &=\varepsilon_\sigma(t)G_\sigma^R(t,t') +\int^{t}_{t^\prime}{{\rm d} \bar{t} }\,\Sigma_\sigma^R(t,\bar{t} ) G_\sigma^R (\bar{t},t^\prime)\nonumber\\ &\quad+\delta(t-t^\prime). \end{align} \tag{ 2 }$$

The on-site interaction is in the mean-field approximation, $\varepsilon_\sigma(t)=\varepsilon_b+U\rho_{\bar\sigma}(t)$, with the spin-independent energy level $\varepsilon_b$ of an isolated atom. The spin ${\bar\sigma}$ is reverse to σ. Both equations have also their conjugate counterparts. As one can see from the lower integration limit, the contact of the bridge atom to leads is switched on at t₀.

The self-energy Σ represents the coupling of the bridge to the leads. It is thus composed of two parts corresponding to the right and the left lead. Its individual elements $\Sigma^{A,R,<}_{\sigma;l,r}$ have equilibrium values given by the energy spectrum on the contact atom of each lead and the bias voltage. These functions are known, independent of the dynamics on the bridge, and depend exclusively on the time difference.

In the approximative treatment one starts from the time diagonal of eq. (1) (plus its conjugate)

$$\begin{eqnarray} \dot{\rho }&= -2\int_{t_0}^{t } {{\rm d} \bar{t} }\,\,{\rm Re}(\Sigma^<(t,\bar{t})G^A(\bar{t},t))\nonumber\\ &\quad -2\int_{t_0}^{t } {{\rm d} \bar{t} }\,\, {\rm Re}(\Sigma^R(t,\bar{t})G^<(\bar{t},t)), \end{eqnarray} \tag{ 3 }$$

which is the precursor equation of the GME. The first (second) term at the right-hand side is the scattering-in (out) integral which describes the tunneling in (out of) the bridge. Since the tunneling self-energy $\Sigma^<$ is determined by electron distribution in the leads and the propagator G^A is evaluated exactly, all approximations we discuss in this paper relate to the scattering-out integral. We suppress the spin indices unless they are necessary for understanding.

Equation (3) is converted to a closed equation for ρ when G^< is approximated by a functional of ρ. A systematic way to such functionals is provided by the so-called reconstruction equation (RE) for $\bar{t} < t$ 

$$\begin{align} G^<(\bar{t},t)&= \rho(\bar{t})G^A(\bar{t},t)\nonumber\\ &\quad+\!\int_{\bar t}^{t}\!{\rm d}\overline{\tau}\left(\Lambda^{\rm in}(\bar{t},\overline{\tau}\,)\!+\!\Lambda^{\rm out}(\bar{t},\overline{\tau})\right)G^A(\,\overline{\tau},t), \end{align} \tag{ 4 }$$

with the leading term $\rho G^A$ and the vertex corrections

$$\begin{equation} \Lambda^{\rm in}(\bar{t},\overline{\tau}\,)=\int^{\bar{t}}_{t_0}{\rm d}t' G^R(\bar{t},t')\Sigma^<(t',\overline{\tau}\,) \end{equation} \tag{ 5 }$$

and

$$\begin{equation} \Lambda^{\rm out}(\bar{t},\overline{\tau}\,)=\int^{\bar{t}}_{t_0}{\rm d}t' G^<(\bar{t},t')\Sigma^A(t',\overline{\tau}\,). \end{equation} \tag{ 6 }$$

Again, there is a conjugate equation for $\bar{t}>t$, which we do not write, since it is not needed in our treatment.

By substitution of the RE (4) into the precursor equation (3) we obtain the kinetic equation

$$\begin{eqnarray} \dot{\rho }&= -2\int_{t_0}^{t } {{\rm d} \bar{t} }\,\,{\rm Re}(\Sigma^<(t,\bar{t})G^A(\bar{t},t))\nonumber\\ &\quad -2\int_{t_0}^{t } {{\rm d} \bar{t} }\,\, {\rm Re}(\Sigma^R(t,\bar{t})\rho(\bar{t})G^A(\bar{t},t))\nonumber \\ &\quad -2\int_{t_0}^{t } {{\rm d} \bar{t} }\,\,{\rm Re}(\Sigma^R(t,\bar{t})\int_{\bar{t}}^{t}{\rm d}{\overline{\tau}}\Lambda^{\rm in}(\bar{t},\overline{\tau})G^A(\overline{\tau},t))\nonumber\\ &\quad -2\int_{t_0}^{t } {{\rm d} \bar{t} }\,\,{\rm Re}(\Sigma^R(t,\bar{t})\int_{\bar{t}}^{t}{\rm d}{\overline{\tau}}\Lambda^{\rm out}(\bar{t},\overline{\tau})G^A(\overline{\tau},t)).\quad \end{eqnarray} \tag{ 7 }$$

Though both vertex corrections originate from the scattering-out, from the operational point of view, the vertex $\Lambda^{\rm in}$ acts as a correction to the scattering-in while the vertex $\Lambda^{\rm out}$ is a correction to scattering-out. Equation (7) is not yet closed, because the vertex $\Lambda^{\rm out}$ is a function of G^<.

The treatment of the vertices $\Lambda^{\rm in,out}$ distinguishes individual approximations discussed in this paper.

• a)  
  The GKBA neglects both vertices, $\Lambda^{\rm in,out}\approx 0$.
• b)  
  The corrected GKBA [13],

  $$\begin{equation} G^<(t,t')=\rho(t)G^A(t,t')+{\cal F}^<_\infty(t-t'), \end{equation} \tag{ 8 }$$

  extends the GKBA by the steady-state correction ${\cal F}^<_\infty=G^<_\infty-\rho_\infty G^A_\infty$. In terms of the Λ functions, this correction reads $\int_{\bar t}^t \mathrm{d}\bar\tau(\Lambda^{\rm in}(\bar t,\bar\tau)+ \Lambda^{\rm out}(\bar t,\bar\tau))G^A(\bar\tau,t)\approx {\cal F}^<_\infty(\bar t-t)$.
• c)  
  The dynamically corrected GKBA uses the steady-state vertex of the tunneling-in, $\Lambda^{\rm in}(\bar{t},\overline{\tau})\approx\Lambda^{\rm in}_\infty(\bar{t}-\overline{\tau})$, and the vertex correction of tunneling-out in the form

  $$\begin{equation} \Lambda^{\rm out}(\bar{t},\overline{\tau})\approx\rho(\bar{t}) \Xi_\infty(\bar{t}-\overline{\tau}), \end{equation} \tag{ 9 }$$

  where

  $$\begin{equation} \Xi(\bar{t},\overline{\tau}\,)=-\int^{\bar{t}}_{t_0}{\rm d}t' G^R(\bar{t},t')\Sigma^A(t',\overline{\tau}\,) \end{equation} \tag{ 10 }$$

  and $\Xi(\bar{t},\overline{\tau}\,)\approx\Xi_\infty(\bar{t}-\overline{\tau}\,)$ is its approximation by the asymptotic steady-state value.
• d)  
  To test sensitivity of the vertex correction to the retardation of the occupation ρ, we evaluate also three extreme cases of the form $\Lambda^{\rm out}(\bar{t},\overline{\tau})\approx\rho(t_c)\Xi_\infty(\bar{t}-\overline{\tau})$, where the time $t_c=\infty,t,\overline{\tau}$ is selected so that the vertex function is numerically feasible.

Our major interest is in the approximation c) first introduced in this paper. To understand its properties we have to make a comparison of all approximations a)–d). The performance of individual approximations is shown in figs. 1 and 2 in contrast to the direct numerical solution of the NEGF equation of motion 1.

a) The apparent failure of the GKBA for this system has been known, and it has stimulated the proposal of the corrected GKBA [13]. The main disagreement with the exact result consists in the incorrect steady-state limit of the GME. In fig. 1 for U = 0, the GME converges to 0.88 while the exact occupation of spin ↑ is 0.73. For $U=0.6$, the GME limit 0.72 is twice larger than the exact value 0.34.

b) The corrected GKBA is a way to impose the correct steady-state limit. Let us show it for non-interacting electrons, $U =0$, when $G^A(\bar{t},t)= G^A_\infty(\bar{t}-t)$ at any times. For t → ∞, the precursor equation (3) yields

$$\begin{eqnarray} & -2\int_{t_0}^{t} {{\rm d} \bar{t} }\,\,{\rm Re}(\Sigma^<(t-\bar{t})G^A_\infty(\bar{t}-t)) \nonumber\\ & -2\int_{t_0}^{t} {{\rm d} \bar{t} }\,\, {\rm Re}(\Sigma^R(t-\bar{t})(\rho_\infty G_\infty^A(\bar{t}-t)\nonumber\\ & +{\cal F}^<_\infty(\bar{t}-t)))\to 0. \end{eqnarray} \tag{ 11 }$$

At large times t → ∞, we subtract eq. (11) from the precursor equation (3) obtaining the relaxation of ρ to $\rho_\infty$,

$$\begin{equation} \dot\rho=-2\int_{t_0}^{t} {{\rm d} \bar{t} }\,\, {\rm Re}(\Sigma^R(t-\bar{t})\left((\rho(\bar{t})-\rho_\infty) G_\infty^A(\bar{t}-t)\right). \end{equation} \tag{ 12 }$$

In figs. 1 and 2 one can see that with the corrected GKBA the occupation $\rho_\uparrow$ converges to $\rho_{\uparrow}^{\infty}$ faster then the exact solution. Briefly, in this approximation the target value is correct but the relaxation rate is too high.

c) In the dynamically corrected GKBA for $G^<_\sigma$, the essential simplification of the vertex correction is given by the time argument of ρ. A straightforward iteration of the RE xxxx4, i.e., the use of the GKBA in the vertex correction, yields $\Lambda^{\rm out}(\bar{t},\overline{\tau}\,)\approx -\int_{t_0}^{\bar{t}}\mathrm{d}t'G^R(\bar{t},t') \rho(t')\Sigma^A(t',\overline{\tau}\,)$. The integration over $t'$ inside time integrals over $\bar{t}$ and $\overline{\tau}$ makes this iteration prohibitively demanding. To reduce the dimension of time integration, we employ the non-retarded Ansatz $G^<(\bar{t},t')\approx -\rho(\bar{t})G^R(\bar{t},t')$ derived below.

Within the non-retarded Ansatz, the remaining time integrals can be simplified. Both vertex corrections (5) and (6) depend on $\rho_{\bar\sigma}$ via the energy level $\varepsilon_\sigma(t)$. We have evaluated $\Lambda^{\rm in}$ and Ξ in the limit $t_0\to -\infty$, when $\Lambda^{\rm in}(\bar{t},\overline{\tau}\,)\to \Lambda^{\rm in}_\infty(\bar{t}-\overline{\tau}\,)$ and $\Xi(\bar{t},\overline{\tau}\,)\to \Xi_\infty(\bar{t}-\overline{\tau}\,)$, using $\varepsilon_\sigma(t)\approx\varepsilon_\sigma(\infty)$. The advanced function G^A in all integrals of the kinetic equation (7) is not approximated, i.e., it is obtained from the Dyson equation (2) with the time-dependent energy level $\varepsilon_\sigma(t)$. To this end both spin components, $\rho_\uparrow$ and $\rho_\downarrow$, are solved together. We present only $\rho_\uparrow$ which reveals larger differences between individual approximations.

Let us motivate the approximation of the vertex $\Lambda^{\rm in}$. Hopjan et al. [16] use the steady-state Green's function as an alternative to the GKBA and show that it works well for the time-dependent tunneling problem. Here we also use the steady-state approximation, but in the vertex corrections only.

In the discussed molecular bridge, the selfenergy Σ represents the embedding not the electron-electron interaction. Accordingly, functions $\Sigma^R(t,t')= \Sigma^R(t-t')$ and $\Sigma^<(t,t')=\Sigma^<(t-t')$ are known, see their Fourier picture in the appendix. The vertex $\Lambda^{\rm in}_\sigma$ depends only on the density of electrons with the reversed spin $\rho_{\bar\sigma}$ via the mean-field energy level $\varepsilon_\sigma(t)=\varepsilon_b+U\rho_{\bar\sigma}(t)$. We have numerically tested this dependency for the on-site interaction $U=0.6$ and found that $\rho_\sigma$ changes only by few percents if we use $\Lambda^{\rm in}_\sigma$ of the non-interacting system. This low sensitivity allows us to use $\rho_{\bar\sigma}(t)\approx\rho_{\bar\sigma}^\infty$, where the $\rho_{\bar\sigma}^\infty$ is either the known steady-state value or the value at large time obtained within iterations. For the constant energy level $\varepsilon_\sigma=\varepsilon_b+U\rho_{\bar\sigma}^\infty$ and the limit $t_0\to-\infty$, the integral 10 depends only on the difference time, $\Xi(\bar{t},\overline{\tau}\,)=\Xi_\infty(\bar{t}-\overline{\tau}\,)$.

d) Ad hoc approximations $\Lambda^{\rm out}(\bar{t},\overline{\tau}\,)\approx\rho(t_c) \Xi_\infty(\bar{t}-\overline{\tau}\,)$ with $t_c=\infty,t,\overline{\tau}$ serve to clarify properties of the vertex correction. Let us start with neglecting the time dependence of ρ using its asymptotic value $\rho_\infty$, i.e., $t_c=\infty$. This approximation leads to the time evolution nearly identical to the corrected GKBA, compare gray and light purple lines in figs. 1 and 2. It documents that the dynamically corrected GKBA differs from the corrected GKBA namely by the dynamics of the occupation.

A more realistic choice is to evaluate the vertex correction at time t from the occupation at the same time, $t_c=t$. The kinetic equation (7) has a form $\dot\rho(t)=\ldots-\rho(t){\cal S}(t)$, where dots stand for the three first terms of (7) and

$$\begin{align} {\cal S}(t)&=2\int_{t_0}^{t } {{\rm d} \bar{t} }\,\,{\rm Re}(\Sigma^R(t-\bar{t})\int_{\bar{t}}^{t}{\rm d}{\overline{\tau}}\Xi_\infty(\bar{t}-\overline{\tau})G^A(\overline{\tau},t)) \end{align} \tag{ 13 }$$

represents the vertex correction to the tunneling out. In fig. 1, for U = 0, this simple and numerically very convenient Markovian choice closely follows the exact solution of NEGF. This extremely good result is fortuitous and appears when the occupation increases in time so that $\rho(t)>\rho(\bar{t})$. The approximation then enhances the vertex correction. For the decreasing occupation the Markovian vertex correction is worse than the dynamically corrected GBKA, see fig. 2.

We have also tested the approximation $\Lambda^{\rm out}(\bar{t},\overline{\tau}\,)\approx\rho(\overline{\tau})\Xi_\infty(\bar{t}-\overline{\tau}\,)$. It requires more numerical effort and leads to a comparably good agreement with the exact result as the above approximation, see figs. 1 and 2. The resulting occupation $\rho_\uparrow$ stays always between the dynamically corrected GKBA and the Markovian vertex correction.

How the choice of time argument t_c affects the kinetic equation can be understood from the individual scattering integrals shown in fig. 3. Both vertex corrections form during the first femtosecond and undergo only minor changes at later times. The $\Lambda^{\rm out}$ is proportional to the occupation $\rho_\uparrow(t_c=\bar{t})$. Using different $t_c >\bar{t}$ leads to an enhanced/reduced $\Lambda^{\rm out}$ for growing/falling occupation of the bridge. The enhancement of $\Lambda^{\rm out}$ correction increases the occupation of the bridge. These trends are confirmed by fig. 2 which shows that for a decreasing occupation, all approximations with $t_c >\bar{t}$ are worse than the dynamically corrected GBKA.

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The non-retarded Ansatz is the zeroth-order approximation of the non-retarded RE. Here we derive this expansion analogous to the RE (4).

For $t >t'$ the function ${\cal G}_\sigma(t,t')=-\theta(t-t')G_\sigma^<(t,t')$ obeys

$$\begin{align} &-{\rm i}\hbar\partial_{t'}{\cal G}_\sigma(t,t')-{\cal G}_\sigma(t,t') \varepsilon_\sigma(t')=\delta(t-t')\rho(t)\nonumber\\ &-\theta(t-t')\int_{t_0}^{t'}{{\rm d} \bar{t} }\,G_\sigma^<(t,\bar{t})\Sigma_\sigma^A (\bar{t},t')\nonumber\\ &-\theta(t-t') \int_{t_0}^{t}{{\rm d} \bar{t} }\,G_\sigma^R(t,\bar{t})\Sigma_\sigma^<(\bar{t},t'). \end{align} \tag{ 14 }$$

We have used eq. (1) to evaluate $\partial_{t'}G^<$. Subtracting ${\cal G}\Sigma^R$ from both sides and multiplying by G^R we arrive at the non-retarded RE

$$\begin{eqnarray} G^<(t,t')&\!=\! -\rho(t)G^R(t,t') \nonumber\\ &\quad +{\rm i}\int_{t'}^{t} {\rm d}\overline{\tau}\int_{t_0}^{t}{{\rm d} \bar{t} }\,G^<(t,\bar{t})\Gamma (\bar{t},\overline{\tau})G^R(\overline{\tau},t')\nonumber\\ &\quad +\!\int_{t'}^{t} {\rm d}\overline{\tau}\int_{t_0}^{t}{{\rm d} \bar{t} }\,G^R(t,\bar{t})\Sigma^<(\bar{t},\overline{\tau})G^R(\overline{\tau},t'),\qquad \end{eqnarray} \tag{ 15 }$$

where $\Gamma={\rm i}(\Sigma^R-\Sigma^A)$.

An approximation of G^< by the first term,

$$\begin{equation} G^<(t,t')=-\rho(t)G^R(t,t'), \end{equation} \tag{ 16 }$$

can be called the non-retarded Ansatz, because the multiplication by G^R from the right hand side does not respect the causal structure of the NEGF: retarded $\times <\times$ advanced. We note that the non-retarded Ansatz thus cannot be obtained by a functional derivative of the causal expansion [20].

We use the non-retarded Ansatz exclusively in the vertex correction. To test its fidelity, we have used (16) to convert the precursor equation into an equation for ρ, and have arrived at a Markovian GME. In figs. 1 and 2 we show that the Markovian GME provides a time evolution very similar to the (retarded) GME, except for quantum interferences at the early stage. This indicated that the non-retarded Ansatz might be a sufficiently good approximation for the internal function of the vertex correction.

We have demonstrated a feasibility of employing approximate vertex corrections to the GKBA in kinetic equations of the GME type. An appreciably improved agreement between the corrected GME and the exact solution obtains even with vertices evaluated from rather crude approximations based on stationary values of propagators. Thanks to these approximations, the vertices increase the computational cost by less than 10% as compared to the (uncorrected) GME.

The dynamically corrected GKBA is derived directly from the complete vertex corrections. Solutions of the corresponding dynamically corrected GME converge to static limits which are much closer to the exact ones than solutions of the (uncorrected) GME. For future implementations, it is essential that the dynamically corrected GKBA does not employ a known steady-state solution. It can be thus applied also to systems which do not evolve towards a steady state. Although our model includes only a single quantum state, the non-retarded RE has a general form valid for any matrix structure of G^<.

The tunneling self-energy of our model $\Sigma^<$ is independent of the GF G^<. In interacting systems, this dependence is crucial and the RE has to be used also for functions in $\Sigma^<[G^<,G^>]$. A full evaluation of the vertex corrections then might be excessively demanding. Perhaps these corrections can be evaluated at selected time instants and used as an estimate of the contributions neglected by GKBA.

The work was supported by project 21-11089S of the Grant Agency of Czech Republic.

Data availability statement: All data that support the findings of this study are included within the article (and any supplementary files).

All components of the self-energy are functions of the time difference, $\Sigma(t,t')=\Sigma(\tau)$, where $\tau=t-t'$. The self-energy represents coupling to the left and the right junction which contribute independently, $\Sigma=\Sigma_L+\Sigma_R$. Individual parts depend on the nickel lead surface density of states (SDOS) via the spectral function of the self-energy

$$\begin{equation} \Delta_{L\sigma}(E)=\lambda_L\cdot\text{SDOS}_{L\sigma}(E), \end{equation} \tag{ A.1 }$$

as

$$\begin{align} \begin{array}{rcl} {\Sigma}_{L\sigma}^{R}(\tau)&=&-{\rm i}\vartheta(\tau)\displaystyle\int\displaystyle\frac{\mathrm{d}E}{2\pi}\Delta_{L\sigma}(E-\mu_L){\rm e}^{-{\rm i} E\tau},\\ {\Sigma}_{L\sigma}^<(\tau)\!&=&{\rm i}\displaystyle\int\displaystyle\frac{\mathrm{d}E}{2\pi}\Delta_{L\sigma}(E-\mu_L)f(E-\mu_L){\rm e}^{-{\rm i} E\tau}. \end{array} \end{align} \tag{ A.2 }$$

Other components are analogous. The Fermi-Dirac distribution in leads f is taken at the room temperature. The junction is biased by the voltage $V=\mu_L-\mu_R$. More about this model the reader can find in [14,21–23].

The failure of the plain GME follows from the rich energy dependency of the local density of state at the bridge $\tilde A$, where the tilde denotes Fourier picture of $A=i(G^R-G^A)$. The $\tilde A$ depends on the position of the bridge energy level with respect to the surface density of states in contacts shown in fig. 4 and the coupling strength $\lambda_{L,R}$, therefore it depends on the spin and the bias, as well as on the occupation of the bridge by electrons of the reversed spin.

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